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Almost sure central limit theorem for partial sums and maxima

✍ Scribed by Peng Zuoxiang; Wang Lili; Saralees Nadarajah

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
107 KB
Volume
282
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let X, X~1~, X~2~, … be i.i.d. random variables with nondegenerate common distribution function F, satisfying EX = 0, EX^2^ = 1. Let X~i~ and M~n~ = max{X~i~, 1 ≀ i ≀ n }. Suppose there exists constants a~n~ > 0, b~n~ ∈ R and a nondegenrate distribution G (y) such that
equation image
Then, we have
equation image
almost surely, where f (x, y) denotes the bounded Lipschitz 1 function and Ξ¦(x) is the standard normal distribution function (Β© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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